02-06-2011, 12:12 PM
Why investigate the factor of safety in the Twin Towers?
There are many reasons why we would want to determine the factor of safety for the steel. The most obvious would be to see if sufficient loads could be applied via load redistribution to fail the (a column)... and cause local failure of the floors around that column.
Another consideration is that the higher the FOS is the less likely is the possibility for a progressive failure to propagate to other columns.
A simple example illustrates the notion of how FOS relates to progressive failure of the columns. In the example let's use 40 columns each carrying identical load with a FOS of 1. If one columns is destroyed or fails for ANY reason... there load it carried is then redistributed to the remaining 39 columns. Before failure each column supported 1/40th of the load and after failure the remaining 39 supported 1/39th of the load. With an FOS each column could ONLY support 1/40 of the load so when a single column fails ALL REMAINING are being loaded PASSED their yield strength and will FAIL from buckling almost immediately. YIKES.
If the FOS was 2 using the same example, each column could support 1/20 of the load. This means that if 19 columns were to fail the remaining 21 would then be supporting 1/21th of the load and the structure would stand. If one more column failed the remain 20 would carry 1/20th of the load and be at their limit. One more column and the remaining 19 would be OVER their limit and fail instantly.
So with an FOS or 1.5 if one column more than 1/3 of the columns fail the remaining columns will buckle immediately.
There are additional factors to consider in progressive column failure scenarios. One is that load redistribution will move to the closest columns to the failed one. And since not all columns ARE of the same load bearing characteristics more information is needed to understand whether and under what conditions will the adjacent columns fail. For example in the twin towers' core the four corner columns were much stronger and carried 2 or 3 times the load of some of the other core columns. So failure of the corner columns would lead to a rapid overloading and failure of adjacent weaker columns.
Look at the attached Column Percentage of Area chart for the core columns on flrs 80-83 (one 36' three story column). Note that the four corner columns contain 4.61% and 4.73% of the core area and carry the same percentage of the core loads. If those 4 columns are "failed" the adjacent columns will immediate see load increases as the loads from the 4 failed columns are redistributed to them. Let's assume the 2 adjacent columns share the new load from the failed columns. They would then carry an additional .5 x 4.61 = 2.3% or .5 x 4.75 = 2.38%. Here are the load percentages for the adjacent columns before and after failure of the corner columns 501 & 1008
502 before failure = 2.81%
502 after failure = 5.31% (it's now loaded at 1.9x its former load)
601 before failure = 1.56%
601 after failure = 3.86% (it's now loaded at 2.5x its former load)
908 before failure = 1.63%
908 after failure = 3% (it's now loaded at 1.8x its former load)
1007 before failure = 2.91%
1007 after failure = 5.28% (it's now loaded at 1.8x its former load)
So in each of these cases IF THE FOS WAS LESS THAN 1.8 the adjacent columns would fail... if they couldn't "offload" some of the additional loads transferred to the by the failure of the corner columns 501 and 1008.
We've seen that the FOS was likely in the 1.65 or lower range. This means that taking out (failure) of the 4 corner columns will lead to the failure (buckling) of the two adjacent columns and then those adjacent to them and so on! TAKE OUT 4 CORNER CORE COLUMNS AND THERE WILL BE A RAPID PROGRESSIVE FAILURE MOVING INWARD TO THE CENTER OF THE CORE AND ALL COLUMNS WILL BUCKLE IF THE FOS IS AROUND 1.65.
It's likely that the adjacent columns CAN redistribute the new loads to adjacent columns. But suppose 4 additional columns were to fail such as 503, 506, 1005 and 1006? If you run the numbers you will see that these 8 column failures with lead to a rapid redistribution and buckling of the entire core... with a FOS of about 1.65.
This understanding was crucial to those who would want to collapse the core at an upper floor so that the floors would have no core side support and drop down onto the intact floors below and initiated the ROOSD.
The ROOSD could be initiated at ANY level, requiring only 4-6 descending floor masses to initiated it. With the steel being thinner the higher you go and column failure from around floor 100 or below would do the trick and require MUCH less incendiaries or explosive energy.
The collapsing core would in addition to pulling the core side of the floors attached to the core down with it, would spring the facade panels or cause them to slip past the lower ones still connected to intact floors with those floors connected to the intact core. The initiation of the ROOSD floor destruction would deliver fractured floors (4-6) onto intact floors when as few as 4-8 key core columns were made to fail.
This is why the FOS is key to understanding of how to engineer the destruction of the towers.
There are many reasons why we would want to determine the factor of safety for the steel. The most obvious would be to see if sufficient loads could be applied via load redistribution to fail the (a column)... and cause local failure of the floors around that column.
Another consideration is that the higher the FOS is the less likely is the possibility for a progressive failure to propagate to other columns.
A simple example illustrates the notion of how FOS relates to progressive failure of the columns. In the example let's use 40 columns each carrying identical load with a FOS of 1. If one columns is destroyed or fails for ANY reason... there load it carried is then redistributed to the remaining 39 columns. Before failure each column supported 1/40th of the load and after failure the remaining 39 supported 1/39th of the load. With an FOS each column could ONLY support 1/40 of the load so when a single column fails ALL REMAINING are being loaded PASSED their yield strength and will FAIL from buckling almost immediately. YIKES.
If the FOS was 2 using the same example, each column could support 1/20 of the load. This means that if 19 columns were to fail the remaining 21 would then be supporting 1/21th of the load and the structure would stand. If one more column failed the remain 20 would carry 1/20th of the load and be at their limit. One more column and the remaining 19 would be OVER their limit and fail instantly.
So with an FOS or 1.5 if one column more than 1/3 of the columns fail the remaining columns will buckle immediately.
There are additional factors to consider in progressive column failure scenarios. One is that load redistribution will move to the closest columns to the failed one. And since not all columns ARE of the same load bearing characteristics more information is needed to understand whether and under what conditions will the adjacent columns fail. For example in the twin towers' core the four corner columns were much stronger and carried 2 or 3 times the load of some of the other core columns. So failure of the corner columns would lead to a rapid overloading and failure of adjacent weaker columns.
Look at the attached Column Percentage of Area chart for the core columns on flrs 80-83 (one 36' three story column). Note that the four corner columns contain 4.61% and 4.73% of the core area and carry the same percentage of the core loads. If those 4 columns are "failed" the adjacent columns will immediate see load increases as the loads from the 4 failed columns are redistributed to them. Let's assume the 2 adjacent columns share the new load from the failed columns. They would then carry an additional .5 x 4.61 = 2.3% or .5 x 4.75 = 2.38%. Here are the load percentages for the adjacent columns before and after failure of the corner columns 501 & 1008
502 before failure = 2.81%
502 after failure = 5.31% (it's now loaded at 1.9x its former load)
601 before failure = 1.56%
601 after failure = 3.86% (it's now loaded at 2.5x its former load)
908 before failure = 1.63%
908 after failure = 3% (it's now loaded at 1.8x its former load)
1007 before failure = 2.91%
1007 after failure = 5.28% (it's now loaded at 1.8x its former load)
So in each of these cases IF THE FOS WAS LESS THAN 1.8 the adjacent columns would fail... if they couldn't "offload" some of the additional loads transferred to the by the failure of the corner columns 501 and 1008.
We've seen that the FOS was likely in the 1.65 or lower range. This means that taking out (failure) of the 4 corner columns will lead to the failure (buckling) of the two adjacent columns and then those adjacent to them and so on! TAKE OUT 4 CORNER CORE COLUMNS AND THERE WILL BE A RAPID PROGRESSIVE FAILURE MOVING INWARD TO THE CENTER OF THE CORE AND ALL COLUMNS WILL BUCKLE IF THE FOS IS AROUND 1.65.
It's likely that the adjacent columns CAN redistribute the new loads to adjacent columns. But suppose 4 additional columns were to fail such as 503, 506, 1005 and 1006? If you run the numbers you will see that these 8 column failures with lead to a rapid redistribution and buckling of the entire core... with a FOS of about 1.65.
This understanding was crucial to those who would want to collapse the core at an upper floor so that the floors would have no core side support and drop down onto the intact floors below and initiated the ROOSD.
The ROOSD could be initiated at ANY level, requiring only 4-6 descending floor masses to initiated it. With the steel being thinner the higher you go and column failure from around floor 100 or below would do the trick and require MUCH less incendiaries or explosive energy.
The collapsing core would in addition to pulling the core side of the floors attached to the core down with it, would spring the facade panels or cause them to slip past the lower ones still connected to intact floors with those floors connected to the intact core. The initiation of the ROOSD floor destruction would deliver fractured floors (4-6) onto intact floors when as few as 4-8 key core columns were made to fail.
This is why the FOS is key to understanding of how to engineer the destruction of the towers.